Abstract
Abstract We introduce a new formulation of the so-called topological recursion, that is defined globally on a compact Riemann surface. We prove that it is equivalent to the generalized recursion for spectral curves with arbitrary ramification. Using this global formulation, we also prove that the correlation functions constructed from the recursion for curves with arbitrary ramification can be obtained as suitable limits of correlation functions for curves with only simple ramification. It then follows that they both satisfy the properties that were originally proved only for curves with simple ramification.
Highlights
The topological recursion proposed in [16] appears in many counting problems in enumerative geometry
Using this global formulation, we prove that the correlation functions constructed from the recursion for curves with arbitrary ramification can be obtained as suitable limits of correlation functions for curves with only simple ramification
For many applications of the topological recursion in enumerative geometry, such as in mirror symmetry, topological string theory and knot theory, spectral curves are given in a form slightly different than what we have studied so far
Summary
The topological recursion proposed in [16] appears in many counting problems in enumerative geometry. We show that this “global” recursion is precisely equivalent to the original and generalized recursions (which we call “local”), which is our main result (Theorem 5) From this theorem, and the global nature of the recursion, it follows that the limits of the correlation functions when ramification points collide are equal to the correlation functions constructed from the limiting curve. The global nature of the recursion, it follows that the limits of the correlation functions when ramification points collide are equal to the correlation functions constructed from the limiting curve Using this main theorem, we prove that the correlation functions constructed for spectral curves with arbitrary ramification satisfy the same properties as the original correlation functions constructed in [16, 18]. In the appendix we prove one of the properties discussed in section 4 explicitly, partly to highlight the fact that it is highly non-trivial that the correlation functions for arbitrary ramification satisfy these properties
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